The Fourier transform of a function h(t) has served as the most important transform in numerous signal processing applications. For example, the Fourier transform is widely used in medical imaging techniques such as Computed Tomography (CT) and Magnetic Resonance Imaging (MRI).
Standard Fourier analysis reveals individual frequency components involved in a signal or an image. However, in many situations of frequencies changing over time or space, the standard Fourier analysis does not provide sufficient information. In numerous applications, frequencies changing over time or space reveal important information. For example, in MRI signal processing, motion caused by respiratory activity, cardiac activity, and blood flow causes temporal changes in a time series signal.
Time-Frequency Representations (TFRs) are capable of localizing spectra of events in time, thus overcoming the deficiency of the standard Fourier analysis and providing a useful tool for signal analysis in numerous applications. However, all TFRs have a substantial disadvantage that hamper or even block their application in present signal processing methods and systems. All TFRs require substantially increased computer memory storage, as well as substantially longer program execution time, compared to the corresponding requirements of the Discrete Fourier Transform (DFT).
One such TFR is the Stockwell-transform (S-transform) disclosed in: Stockwell R. G., Mansinha L., Lowe R. P., “Localization of the complex spectrum: the S-transform”, IEEE Trans. Signal Process, 1996; 44, 998-1001. The S-transform is a spectral localization transform that uses a frequency adapted Gaussian window to achieve optimum resolution at each frequency. Employment of the S-transform is highly beneficial in numerous applications. For example, a one-dimensional version of the S-transform applied to time series signal data, such as time course functional MRI (fMRI) data, enables localizing and removing of noise components and artifacts, while a two dimensional version of the S-transform provides local textural information for each point in an image. This gives a texture map of an MRI image that enhances differences, indicating lesions or other abnormalities due to disease activity that are difficult to distinguish in conventional MR images. Unfortunately, the system requirements of the S-transform regarding computer memory storage, as well as execution time, prohibit employment of this highly beneficial signal processing tool in a clinical setting.
It would be advantageous to overcome the drawbacks of the S-transform by providing a signal processing method and system, based on the S-transform but having substantially reduced system requirements. Such a signal processing method and system based on the S-transform would be highly beneficial, not only for signal processing in medical applications, but in numerous other industrial and scientific applications.